Termination w.r.t. Q proof of /tmp/aprove_benchmark_example

Termination w.r.t. Q of the following Term Rewriting System could be proven:

Q restricted rewrite system:
The TRS R consists of the following rules:

le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
zero(0) → true
zero(s(x)) → false
id(0) → 0
id(s(x)) → s(id(x))
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
mod(x, y) → if_mod(zero(x), zero(y), le(y, x), id(x), id(y))
if_mod(true, b1, b2, x, y) → 0
if_mod(false, b1, b2, x, y) → if2(b1, b2, x, y)
if2(true, b2, x, y) → 0
if2(false, b2, x, y) → if3(b2, x, y)
if3(true, x, y) → mod(minus(x, y), s(y))
if3(false, x, y) → x

Q is empty.

↳ QTRS

  ↳ Overlay + Local Confluence

Q restricted rewrite system:
The TRS R consists of the following rules:

le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
zero(0) → true
zero(s(x)) → false
id(0) → 0
id(s(x)) → s(id(x))
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
mod(x, y) → if_mod(zero(x), zero(y), le(y, x), id(x), id(y))
if_mod(true, b1, b2, x, y) → 0
if_mod(false, b1, b2, x, y) → if2(b1, b2, x, y)
if2(true, b2, x, y) → 0
if2(false, b2, x, y) → if3(b2, x, y)
if3(true, x, y) → mod(minus(x, y), s(y))
if3(false, x, y) → x

Q is empty.

The TRS is overlay and locally confluent. By [19] we can switch to innermost.

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

Q restricted rewrite system:
The TRS R consists of the following rules:

le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
zero(0) → true
zero(s(x)) → false
id(0) → 0
id(s(x)) → s(id(x))
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
mod(x, y) → if_mod(zero(x), zero(y), le(y, x), id(x), id(y))
if_mod(true, b1, b2, x, y) → 0
if_mod(false, b1, b2, x, y) → if2(b1, b2, x, y)
if2(true, b2, x, y) → 0
if2(false, b2, x, y) → if3(b2, x, y)
if3(true, x, y) → mod(minus(x, y), s(y))
if3(false, x, y) → x

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
zero(0)
zero(s(x0))
id(0)
id(s(x0))
minus(x0, 0)
minus(s(x0), s(x1))
mod(x0, x1)
if_mod(true, x0, x1, x2, x3)
if_mod(false, x0, x1, x2, x3)
if2(true, x0, x1, x2)
if2(false, x0, x1, x2)
if3(true, x0, x1)
if3(false, x0, x1)

Using Dependency Pairs [1,15] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

IF3(true, x, y) → MINUS(x, y)
MOD(x, y) → ID(y)
ID(s(x)) → ID(x)
MOD(x, y) → ZERO(x)
LE(s(x), s(y)) → LE(x, y)
MOD(x, y) → LE(y, x)
IF3(true, x, y) → MOD(minus(x, y), s(y))
MINUS(s(x), s(y)) → MINUS(x, y)
IF2(false, b2, x, y) → IF3(b2, x, y)
IF_MOD(false, b1, b2, x, y) → IF2(b1, b2, x, y)
MOD(x, y) → ID(x)
MOD(x, y) → IF_MOD(zero(x), zero(y), le(y, x), id(x), id(y))
MOD(x, y) → ZERO(y)

The TRS R consists of the following rules:

le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
zero(0) → true
zero(s(x)) → false
id(0) → 0
id(s(x)) → s(id(x))
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
mod(x, y) → if_mod(zero(x), zero(y), le(y, x), id(x), id(y))
if_mod(true, b1, b2, x, y) → 0
if_mod(false, b1, b2, x, y) → if2(b1, b2, x, y)
if2(true, b2, x, y) → 0
if2(false, b2, x, y) → if3(b2, x, y)
if3(true, x, y) → mod(minus(x, y), s(y))
if3(false, x, y) → x

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
zero(0)
zero(s(x0))
id(0)
id(s(x0))
minus(x0, 0)
minus(s(x0), s(x1))
mod(x0, x1)
if_mod(true, x0, x1, x2, x3)
if_mod(false, x0, x1, x2, x3)
if2(true, x0, x1, x2)
if2(false, x0, x1, x2)
if3(true, x0, x1)
if3(false, x0, x1)

We have to consider all minimal (P,Q,R)-chains.

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

IF3(true, x, y) → MINUS(x, y)
MOD(x, y) → ID(y)
ID(s(x)) → ID(x)
MOD(x, y) → ZERO(x)
LE(s(x), s(y)) → LE(x, y)
MOD(x, y) → LE(y, x)
IF3(true, x, y) → MOD(minus(x, y), s(y))
MINUS(s(x), s(y)) → MINUS(x, y)
IF2(false, b2, x, y) → IF3(b2, x, y)
IF_MOD(false, b1, b2, x, y) → IF2(b1, b2, x, y)
MOD(x, y) → ID(x)
MOD(x, y) → IF_MOD(zero(x), zero(y), le(y, x), id(x), id(y))
MOD(x, y) → ZERO(y)

The TRS R consists of the following rules:

le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
zero(0) → true
zero(s(x)) → false
id(0) → 0
id(s(x)) → s(id(x))
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
mod(x, y) → if_mod(zero(x), zero(y), le(y, x), id(x), id(y))
if_mod(true, b1, b2, x, y) → 0
if_mod(false, b1, b2, x, y) → if2(b1, b2, x, y)
if2(true, b2, x, y) → 0
if2(false, b2, x, y) → if3(b2, x, y)
if3(true, x, y) → mod(minus(x, y), s(y))
if3(false, x, y) → x

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
zero(0)
zero(s(x0))
id(0)
id(s(x0))
minus(x0, 0)
minus(s(x0), s(x1))
mod(x0, x1)
if_mod(true, x0, x1, x2, x3)
if_mod(false, x0, x1, x2, x3)
if2(true, x0, x1, x2)
if2(false, x0, x1, x2)
if3(true, x0, x1)
if3(false, x0, x1)

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 4 SCCs with 6 less nodes.

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ UsableRulesProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

MINUS(s(x), s(y)) → MINUS(x, y)

The TRS R consists of the following rules:

le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
zero(0) → true
zero(s(x)) → false
id(0) → 0
id(s(x)) → s(id(x))
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
mod(x, y) → if_mod(zero(x), zero(y), le(y, x), id(x), id(y))
if_mod(true, b1, b2, x, y) → 0
if_mod(false, b1, b2, x, y) → if2(b1, b2, x, y)
if2(true, b2, x, y) → 0
if2(false, b2, x, y) → if3(b2, x, y)
if3(true, x, y) → mod(minus(x, y), s(y))
if3(false, x, y) → x

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
zero(0)
zero(s(x0))
id(0)
id(s(x0))
minus(x0, 0)
minus(s(x0), s(x1))
mod(x0, x1)
if_mod(true, x0, x1, x2, x3)
if_mod(false, x0, x1, x2, x3)
if2(true, x0, x1, x2)
if2(false, x0, x1, x2)
if3(true, x0, x1)
if3(false, x0, x1)

We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

MINUS(s(x), s(y)) → MINUS(x, y)

R is empty.
The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
zero(0)
zero(s(x0))
id(0)
id(s(x0))
minus(x0, 0)
minus(s(x0), s(x1))
mod(x0, x1)
if_mod(true, x0, x1, x2, x3)
if_mod(false, x0, x1, x2, x3)
if2(true, x0, x1, x2)
if2(false, x0, x1, x2)
if3(true, x0, x1)
if3(false, x0, x1)

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
zero(0)
zero(s(x0))
id(0)
id(s(x0))
minus(x0, 0)
minus(s(x0), s(x1))
mod(x0, x1)
if_mod(true, x0, x1, x2, x3)
if_mod(false, x0, x1, x2, x3)
if2(true, x0, x1, x2)
if2(false, x0, x1, x2)
if3(true, x0, x1)
if3(false, x0, x1)

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

MINUS(s(x), s(y)) → MINUS(x, y)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

MINUS(s(x), s(y)) → MINUS(x, y)
The graph contains the following edges 1 > 1, 2 > 2

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

              ↳ QDP

              ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

ID(s(x)) → ID(x)

The TRS R consists of the following rules:

le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
zero(0) → true
zero(s(x)) → false
id(0) → 0
id(s(x)) → s(id(x))
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
mod(x, y) → if_mod(zero(x), zero(y), le(y, x), id(x), id(y))
if_mod(true, b1, b2, x, y) → 0
if_mod(false, b1, b2, x, y) → if2(b1, b2, x, y)
if2(true, b2, x, y) → 0
if2(false, b2, x, y) → if3(b2, x, y)
if3(true, x, y) → mod(minus(x, y), s(y))
if3(false, x, y) → x

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
zero(0)
zero(s(x0))
id(0)
id(s(x0))
minus(x0, 0)
minus(s(x0), s(x1))
mod(x0, x1)
if_mod(true, x0, x1, x2, x3)
if_mod(false, x0, x1, x2, x3)
if2(true, x0, x1, x2)
if2(false, x0, x1, x2)
if3(true, x0, x1)
if3(false, x0, x1)

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

              ↳ QDP

              ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

ID(s(x)) → ID(x)

R is empty.
The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
zero(0)
zero(s(x0))
id(0)
id(s(x0))
minus(x0, 0)
minus(s(x0), s(x1))
mod(x0, x1)
if_mod(true, x0, x1, x2, x3)
if_mod(false, x0, x1, x2, x3)
if2(true, x0, x1, x2)
if2(false, x0, x1, x2)
if3(true, x0, x1)
if3(false, x0, x1)

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
zero(0)
zero(s(x0))
id(0)
id(s(x0))
minus(x0, 0)
minus(s(x0), s(x1))
mod(x0, x1)
if_mod(true, x0, x1, x2, x3)
if_mod(false, x0, x1, x2, x3)
if2(true, x0, x1, x2)
if2(false, x0, x1, x2)
if3(true, x0, x1)
if3(false, x0, x1)

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ QDP

              ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

ID(s(x)) → ID(x)

From the DPs we obtained the following set of size-change graphs:

ID(s(x)) → ID(x)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

              ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

LE(s(x), s(y)) → LE(x, y)

The TRS R consists of the following rules:

le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
zero(0) → true
zero(s(x)) → false
id(0) → 0
id(s(x)) → s(id(x))
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
mod(x, y) → if_mod(zero(x), zero(y), le(y, x), id(x), id(y))
if_mod(true, b1, b2, x, y) → 0
if_mod(false, b1, b2, x, y) → if2(b1, b2, x, y)
if2(true, b2, x, y) → 0
if2(false, b2, x, y) → if3(b2, x, y)
if3(true, x, y) → mod(minus(x, y), s(y))
if3(false, x, y) → x

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
zero(0)
zero(s(x0))
id(0)
id(s(x0))
minus(x0, 0)
minus(s(x0), s(x1))
mod(x0, x1)
if_mod(true, x0, x1, x2, x3)
if_mod(false, x0, x1, x2, x3)
if2(true, x0, x1, x2)
if2(false, x0, x1, x2)
if3(true, x0, x1)
if3(false, x0, x1)

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

              ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

LE(s(x), s(y)) → LE(x, y)

R is empty.
The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
zero(0)
zero(s(x0))
id(0)
id(s(x0))
minus(x0, 0)
minus(s(x0), s(x1))
mod(x0, x1)
if_mod(true, x0, x1, x2, x3)
if_mod(false, x0, x1, x2, x3)
if2(true, x0, x1, x2)
if2(false, x0, x1, x2)
if3(true, x0, x1)
if3(false, x0, x1)

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
zero(0)
zero(s(x0))
id(0)
id(s(x0))
minus(x0, 0)
minus(s(x0), s(x1))
mod(x0, x1)
if_mod(true, x0, x1, x2, x3)
if_mod(false, x0, x1, x2, x3)
if2(true, x0, x1, x2)
if2(false, x0, x1, x2)
if3(true, x0, x1)
if3(false, x0, x1)

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

LE(s(x), s(y)) → LE(x, y)

From the DPs we obtained the following set of size-change graphs:

LE(s(x), s(y)) → LE(x, y)
The graph contains the following edges 1 > 1, 2 > 2

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

Q DP problem:
The TRS P consists of the following rules:

IF3(true, x, y) → MOD(minus(x, y), s(y))
IF2(false, b2, x, y) → IF3(b2, x, y)
IF_MOD(false, b1, b2, x, y) → IF2(b1, b2, x, y)
MOD(x, y) → IF_MOD(zero(x), zero(y), le(y, x), id(x), id(y))

The TRS R consists of the following rules:

le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
zero(0) → true
zero(s(x)) → false
id(0) → 0
id(s(x)) → s(id(x))
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
mod(x, y) → if_mod(zero(x), zero(y), le(y, x), id(x), id(y))
if_mod(true, b1, b2, x, y) → 0
if_mod(false, b1, b2, x, y) → if2(b1, b2, x, y)
if2(true, b2, x, y) → 0
if2(false, b2, x, y) → if3(b2, x, y)
if3(true, x, y) → mod(minus(x, y), s(y))
if3(false, x, y) → x

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
zero(0)
zero(s(x0))
id(0)
id(s(x0))
minus(x0, 0)
minus(s(x0), s(x1))
mod(x0, x1)
if_mod(true, x0, x1, x2, x3)
if_mod(false, x0, x1, x2, x3)
if2(true, x0, x1, x2)
if2(false, x0, x1, x2)
if3(true, x0, x1)
if3(false, x0, x1)

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

Q DP problem:
The TRS P consists of the following rules:

IF3(true, x, y) → MOD(minus(x, y), s(y))
IF2(false, b2, x, y) → IF3(b2, x, y)
IF_MOD(false, b1, b2, x, y) → IF2(b1, b2, x, y)
MOD(x, y) → IF_MOD(zero(x), zero(y), le(y, x), id(x), id(y))

The TRS R consists of the following rules:

minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
zero(0) → true
zero(s(x)) → false
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
id(0) → 0
id(s(x)) → s(id(x))

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
zero(0)
zero(s(x0))
id(0)
id(s(x0))
minus(x0, 0)
minus(s(x0), s(x1))
mod(x0, x1)
if_mod(true, x0, x1, x2, x3)
if_mod(false, x0, x1, x2, x3)
if2(true, x0, x1, x2)
if2(false, x0, x1, x2)
if3(true, x0, x1)
if3(false, x0, x1)

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.

mod(x0, x1)
if_mod(true, x0, x1, x2, x3)
if_mod(false, x0, x1, x2, x3)
if2(true, x0, x1, x2)
if2(false, x0, x1, x2)
if3(true, x0, x1)
if3(false, x0, x1)

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

Q DP problem:
The TRS P consists of the following rules:

IF3(true, x, y) → MOD(minus(x, y), s(y))
IF2(false, b2, x, y) → IF3(b2, x, y)
IF_MOD(false, b1, b2, x, y) → IF2(b1, b2, x, y)
MOD(x, y) → IF_MOD(zero(x), zero(y), le(y, x), id(x), id(y))

The TRS R consists of the following rules:

minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
zero(0) → true
zero(s(x)) → false
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
id(0) → 0
id(s(x)) → s(id(x))

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
zero(0)
zero(s(x0))
id(0)
id(s(x0))
minus(x0, 0)
minus(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MOD(x, y) → IF_MOD(zero(x), zero(y), le(y, x), id(x), id(y)) at position [0] we obtained the following new rules:

MOD(s(x0), y1) → IF_MOD(false, zero(y1), le(y1, s(x0)), id(s(x0)), id(y1))
MOD(0, y1) → IF_MOD(true, zero(y1), le(y1, 0), id(0), id(y1))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

IF3(true, x, y) → MOD(minus(x, y), s(y))
MOD(0, y1) → IF_MOD(true, zero(y1), le(y1, 0), id(0), id(y1))
IF2(false, b2, x, y) → IF3(b2, x, y)
IF_MOD(false, b1, b2, x, y) → IF2(b1, b2, x, y)
MOD(s(x0), y1) → IF_MOD(false, zero(y1), le(y1, s(x0)), id(s(x0)), id(y1))

The TRS R consists of the following rules:

minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
zero(0) → true
zero(s(x)) → false
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
id(0) → 0
id(s(x)) → s(id(x))

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
zero(0)
zero(s(x0))
id(0)
id(s(x0))
minus(x0, 0)
minus(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

Q DP problem:
The TRS P consists of the following rules:

IF3(true, x, y) → MOD(minus(x, y), s(y))
IF2(false, b2, x, y) → IF3(b2, x, y)
IF_MOD(false, b1, b2, x, y) → IF2(b1, b2, x, y)
MOD(s(x0), y1) → IF_MOD(false, zero(y1), le(y1, s(x0)), id(s(x0)), id(y1))

The TRS R consists of the following rules:

minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
zero(0) → true
zero(s(x)) → false
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
id(0) → 0
id(s(x)) → s(id(x))

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
zero(0)
zero(s(x0))
id(0)
id(s(x0))
minus(x0, 0)
minus(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule MOD(s(x0), y1) → IF_MOD(false, zero(y1), le(y1, s(x0)), id(s(x0)), id(y1)) at position [3] we obtained the following new rules:

MOD(s(x0), y1) → IF_MOD(false, zero(y1), le(y1, s(x0)), s(id(x0)), id(y1))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Narrowing

Q DP problem:
The TRS P consists of the following rules:

MOD(s(x0), y1) → IF_MOD(false, zero(y1), le(y1, s(x0)), s(id(x0)), id(y1))
IF3(true, x, y) → MOD(minus(x, y), s(y))
IF2(false, b2, x, y) → IF3(b2, x, y)
IF_MOD(false, b1, b2, x, y) → IF2(b1, b2, x, y)

The TRS R consists of the following rules:

minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
zero(0) → true
zero(s(x)) → false
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
id(0) → 0
id(s(x)) → s(id(x))

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
zero(0)
zero(s(x0))
id(0)
id(s(x0))
minus(x0, 0)
minus(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule IF3(true, x, y) → MOD(minus(x, y), s(y)) at position [0] we obtained the following new rules:

IF3(true, s(x0), s(x1)) → MOD(minus(x0, x1), s(s(x1)))
IF3(true, x0, 0) → MOD(x0, s(0))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Instantiation

Q DP problem:
The TRS P consists of the following rules:

MOD(s(x0), y1) → IF_MOD(false, zero(y1), le(y1, s(x0)), s(id(x0)), id(y1))
IF3(true, s(x0), s(x1)) → MOD(minus(x0, x1), s(s(x1)))
IF2(false, b2, x, y) → IF3(b2, x, y)
IF_MOD(false, b1, b2, x, y) → IF2(b1, b2, x, y)
IF3(true, x0, 0) → MOD(x0, s(0))

The TRS R consists of the following rules:

minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
zero(0) → true
zero(s(x)) → false
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
id(0) → 0
id(s(x)) → s(id(x))

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
zero(0)
zero(s(x0))
id(0)
id(s(x0))
minus(x0, 0)
minus(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule IF_MOD(false, b1, b2, x, y) → IF2(b1, b2, x, y) we obtained the following new rules:

IF_MOD(false, y_0, y_1, s(y_2), y_3) → IF2(y_0, y_1, s(y_2), y_3)

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Instantiation

                                          ↳ QDP

                                            ↳ Instantiation

Q DP problem:
The TRS P consists of the following rules:

MOD(s(x0), y1) → IF_MOD(false, zero(y1), le(y1, s(x0)), s(id(x0)), id(y1))
IF3(true, s(x0), s(x1)) → MOD(minus(x0, x1), s(s(x1)))
IF2(false, b2, x, y) → IF3(b2, x, y)
IF3(true, x0, 0) → MOD(x0, s(0))
IF_MOD(false, y_0, y_1, s(y_2), y_3) → IF2(y_0, y_1, s(y_2), y_3)

The TRS R consists of the following rules:

minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
zero(0) → true
zero(s(x)) → false
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
id(0) → 0
id(s(x)) → s(id(x))

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
zero(0)
zero(s(x0))
id(0)
id(s(x0))
minus(x0, 0)
minus(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule IF2(false, b2, x, y) → IF3(b2, x, y) we obtained the following new rules:

IF2(false, z1, s(z2), z3) → IF3(z1, s(z2), z3)

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Instantiation

                                          ↳ QDP

                                            ↳ Instantiation

                                              ↳ QDP

                                                ↳ Instantiation

Q DP problem:
The TRS P consists of the following rules:

IF2(false, z1, s(z2), z3) → IF3(z1, s(z2), z3)
MOD(s(x0), y1) → IF_MOD(false, zero(y1), le(y1, s(x0)), s(id(x0)), id(y1))
IF3(true, s(x0), s(x1)) → MOD(minus(x0, x1), s(s(x1)))
IF3(true, x0, 0) → MOD(x0, s(0))
IF_MOD(false, y_0, y_1, s(y_2), y_3) → IF2(y_0, y_1, s(y_2), y_3)

The TRS R consists of the following rules:

minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
zero(0) → true
zero(s(x)) → false
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
id(0) → 0
id(s(x)) → s(id(x))

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
zero(0)
zero(s(x0))
id(0)
id(s(x0))
minus(x0, 0)
minus(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule MOD(s(x0), y1) → IF_MOD(false, zero(y1), le(y1, s(x0)), s(id(x0)), id(y1)) we obtained the following new rules:

MOD(s(x0), s(0)) → IF_MOD(false, zero(s(0)), le(s(0), s(x0)), s(id(x0)), id(s(0)))
MOD(s(x0), s(s(z1))) → IF_MOD(false, zero(s(s(z1))), le(s(s(z1)), s(x0)), s(id(x0)), id(s(s(z1))))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Instantiation

                                          ↳ QDP

                                            ↳ Instantiation

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

Q DP problem:
The TRS P consists of the following rules:

IF2(false, z1, s(z2), z3) → IF3(z1, s(z2), z3)
IF3(true, s(x0), s(x1)) → MOD(minus(x0, x1), s(s(x1)))
MOD(s(x0), s(s(z1))) → IF_MOD(false, zero(s(s(z1))), le(s(s(z1)), s(x0)), s(id(x0)), id(s(s(z1))))
MOD(s(x0), s(0)) → IF_MOD(false, zero(s(0)), le(s(0), s(x0)), s(id(x0)), id(s(0)))
IF3(true, x0, 0) → MOD(x0, s(0))
IF_MOD(false, y_0, y_1, s(y_2), y_3) → IF2(y_0, y_1, s(y_2), y_3)

The TRS R consists of the following rules:

minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
zero(0) → true
zero(s(x)) → false
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
id(0) → 0
id(s(x)) → s(id(x))

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
zero(0)
zero(s(x0))
id(0)
id(s(x0))
minus(x0, 0)
minus(s(x0), s(x1))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Instantiation

                                          ↳ QDP

                                            ↳ Instantiation

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ Rewriting

Q DP problem:
The TRS P consists of the following rules:

IF2(false, z1, s(z2), z3) → IF3(z1, s(z2), z3)
IF3(true, s(x0), s(x1)) → MOD(minus(x0, x1), s(s(x1)))
MOD(s(x0), s(s(z1))) → IF_MOD(false, zero(s(s(z1))), le(s(s(z1)), s(x0)), s(id(x0)), id(s(s(z1))))
MOD(s(x0), s(0)) → IF_MOD(false, zero(s(0)), le(s(0), s(x0)), s(id(x0)), id(s(0)))
IF3(true, x0, 0) → MOD(x0, s(0))
IF_MOD(false, y_0, y_1, s(y_2), y_3) → IF2(y_0, y_1, s(y_2), y_3)

The TRS R consists of the following rules:

minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
zero(s(x)) → false
le(s(x), s(y)) → le(x, y)
id(0) → 0
id(s(x)) → s(id(x))
le(0, y) → true
le(s(x), 0) → false

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
zero(0)
zero(s(x0))
id(0)
id(s(x0))
minus(x0, 0)
minus(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule MOD(s(x0), s(0)) → IF_MOD(false, zero(s(0)), le(s(0), s(x0)), s(id(x0)), id(s(0))) at position [1] we obtained the following new rules:

MOD(s(x0), s(0)) → IF_MOD(false, false, le(s(0), s(x0)), s(id(x0)), id(s(0)))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Instantiation

                                          ↳ QDP

                                            ↳ Instantiation

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

Q DP problem:
The TRS P consists of the following rules:

IF2(false, z1, s(z2), z3) → IF3(z1, s(z2), z3)
IF3(true, s(x0), s(x1)) → MOD(minus(x0, x1), s(s(x1)))
MOD(s(x0), s(s(z1))) → IF_MOD(false, zero(s(s(z1))), le(s(s(z1)), s(x0)), s(id(x0)), id(s(s(z1))))
MOD(s(x0), s(0)) → IF_MOD(false, false, le(s(0), s(x0)), s(id(x0)), id(s(0)))
IF3(true, x0, 0) → MOD(x0, s(0))
IF_MOD(false, y_0, y_1, s(y_2), y_3) → IF2(y_0, y_1, s(y_2), y_3)

The TRS R consists of the following rules:

minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
zero(s(x)) → false
le(s(x), s(y)) → le(x, y)
id(0) → 0
id(s(x)) → s(id(x))
le(0, y) → true
le(s(x), 0) → false

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
zero(0)
zero(s(x0))
id(0)
id(s(x0))
minus(x0, 0)
minus(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule MOD(s(x0), s(s(z1))) → IF_MOD(false, zero(s(s(z1))), le(s(s(z1)), s(x0)), s(id(x0)), id(s(s(z1)))) at position [1] we obtained the following new rules:

MOD(s(x0), s(s(z1))) → IF_MOD(false, false, le(s(s(z1)), s(x0)), s(id(x0)), id(s(s(z1))))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Instantiation

                                          ↳ QDP

                                            ↳ Instantiation

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

Q DP problem:
The TRS P consists of the following rules:

IF2(false, z1, s(z2), z3) → IF3(z1, s(z2), z3)
IF3(true, s(x0), s(x1)) → MOD(minus(x0, x1), s(s(x1)))
MOD(s(x0), s(0)) → IF_MOD(false, false, le(s(0), s(x0)), s(id(x0)), id(s(0)))
MOD(s(x0), s(s(z1))) → IF_MOD(false, false, le(s(s(z1)), s(x0)), s(id(x0)), id(s(s(z1))))
IF3(true, x0, 0) → MOD(x0, s(0))
IF_MOD(false, y_0, y_1, s(y_2), y_3) → IF2(y_0, y_1, s(y_2), y_3)

The TRS R consists of the following rules:

minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
zero(s(x)) → false
le(s(x), s(y)) → le(x, y)
id(0) → 0
id(s(x)) → s(id(x))
le(0, y) → true
le(s(x), 0) → false

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
zero(0)
zero(s(x0))
id(0)
id(s(x0))
minus(x0, 0)
minus(s(x0), s(x1))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Instantiation

                                          ↳ QDP

                                            ↳ Instantiation

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

Q DP problem:
The TRS P consists of the following rules:

IF2(false, z1, s(z2), z3) → IF3(z1, s(z2), z3)
IF3(true, s(x0), s(x1)) → MOD(minus(x0, x1), s(s(x1)))
MOD(s(x0), s(0)) → IF_MOD(false, false, le(s(0), s(x0)), s(id(x0)), id(s(0)))
MOD(s(x0), s(s(z1))) → IF_MOD(false, false, le(s(s(z1)), s(x0)), s(id(x0)), id(s(s(z1))))
IF3(true, x0, 0) → MOD(x0, s(0))
IF_MOD(false, y_0, y_1, s(y_2), y_3) → IF2(y_0, y_1, s(y_2), y_3)

The TRS R consists of the following rules:

minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
le(s(x), s(y)) → le(x, y)
id(0) → 0
id(s(x)) → s(id(x))
le(0, y) → true
le(s(x), 0) → false

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
zero(0)
zero(s(x0))
id(0)
id(s(x0))
minus(x0, 0)
minus(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.

zero(0)
zero(s(x0))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Instantiation

                                          ↳ QDP

                                            ↳ Instantiation

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

Q DP problem:
The TRS P consists of the following rules:

IF2(false, z1, s(z2), z3) → IF3(z1, s(z2), z3)
IF3(true, s(x0), s(x1)) → MOD(minus(x0, x1), s(s(x1)))
MOD(s(x0), s(0)) → IF_MOD(false, false, le(s(0), s(x0)), s(id(x0)), id(s(0)))
MOD(s(x0), s(s(z1))) → IF_MOD(false, false, le(s(s(z1)), s(x0)), s(id(x0)), id(s(s(z1))))
IF3(true, x0, 0) → MOD(x0, s(0))
IF_MOD(false, y_0, y_1, s(y_2), y_3) → IF2(y_0, y_1, s(y_2), y_3)

The TRS R consists of the following rules:

minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
le(s(x), s(y)) → le(x, y)
id(0) → 0
id(s(x)) → s(id(x))
le(0, y) → true
le(s(x), 0) → false

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
id(0)
id(s(x0))
minus(x0, 0)
minus(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule MOD(s(x0), s(0)) → IF_MOD(false, false, le(s(0), s(x0)), s(id(x0)), id(s(0))) at position [2] we obtained the following new rules:

MOD(s(x0), s(0)) → IF_MOD(false, false, le(0, x0), s(id(x0)), id(s(0)))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Instantiation

                                          ↳ QDP

                                            ↳ Instantiation

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

Q DP problem:
The TRS P consists of the following rules:

IF2(false, z1, s(z2), z3) → IF3(z1, s(z2), z3)
IF3(true, s(x0), s(x1)) → MOD(minus(x0, x1), s(s(x1)))
MOD(s(x0), s(s(z1))) → IF_MOD(false, false, le(s(s(z1)), s(x0)), s(id(x0)), id(s(s(z1))))
MOD(s(x0), s(0)) → IF_MOD(false, false, le(0, x0), s(id(x0)), id(s(0)))
IF3(true, x0, 0) → MOD(x0, s(0))
IF_MOD(false, y_0, y_1, s(y_2), y_3) → IF2(y_0, y_1, s(y_2), y_3)

The TRS R consists of the following rules:

minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
le(s(x), s(y)) → le(x, y)
id(0) → 0
id(s(x)) → s(id(x))
le(0, y) → true
le(s(x), 0) → false

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
id(0)
id(s(x0))
minus(x0, 0)
minus(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule MOD(s(x0), s(s(z1))) → IF_MOD(false, false, le(s(s(z1)), s(x0)), s(id(x0)), id(s(s(z1)))) at position [2] we obtained the following new rules:

MOD(s(x0), s(s(z1))) → IF_MOD(false, false, le(s(z1), x0), s(id(x0)), id(s(s(z1))))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Instantiation

                                          ↳ QDP

                                            ↳ Instantiation

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

Q DP problem:
The TRS P consists of the following rules:

IF2(false, z1, s(z2), z3) → IF3(z1, s(z2), z3)
IF3(true, s(x0), s(x1)) → MOD(minus(x0, x1), s(s(x1)))
MOD(s(x0), s(0)) → IF_MOD(false, false, le(0, x0), s(id(x0)), id(s(0)))
IF3(true, x0, 0) → MOD(x0, s(0))
MOD(s(x0), s(s(z1))) → IF_MOD(false, false, le(s(z1), x0), s(id(x0)), id(s(s(z1))))
IF_MOD(false, y_0, y_1, s(y_2), y_3) → IF2(y_0, y_1, s(y_2), y_3)

The TRS R consists of the following rules:

minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
le(s(x), s(y)) → le(x, y)
id(0) → 0
id(s(x)) → s(id(x))
le(0, y) → true
le(s(x), 0) → false

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
id(0)
id(s(x0))
minus(x0, 0)
minus(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule MOD(s(x0), s(0)) → IF_MOD(false, false, le(0, x0), s(id(x0)), id(s(0))) at position [2] we obtained the following new rules:

MOD(s(x0), s(0)) → IF_MOD(false, false, true, s(id(x0)), id(s(0)))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Instantiation

                                          ↳ QDP

                                            ↳ Instantiation

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

Q DP problem:
The TRS P consists of the following rules:

IF2(false, z1, s(z2), z3) → IF3(z1, s(z2), z3)
IF3(true, s(x0), s(x1)) → MOD(minus(x0, x1), s(s(x1)))
MOD(s(x0), s(0)) → IF_MOD(false, false, true, s(id(x0)), id(s(0)))
IF3(true, x0, 0) → MOD(x0, s(0))
MOD(s(x0), s(s(z1))) → IF_MOD(false, false, le(s(z1), x0), s(id(x0)), id(s(s(z1))))
IF_MOD(false, y_0, y_1, s(y_2), y_3) → IF2(y_0, y_1, s(y_2), y_3)

The TRS R consists of the following rules:

minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
le(s(x), s(y)) → le(x, y)
id(0) → 0
id(s(x)) → s(id(x))
le(0, y) → true
le(s(x), 0) → false

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
id(0)
id(s(x0))
minus(x0, 0)
minus(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule MOD(s(x0), s(s(z1))) → IF_MOD(false, false, le(s(z1), x0), s(id(x0)), id(s(s(z1)))) at position [4] we obtained the following new rules:

MOD(s(x0), s(s(z1))) → IF_MOD(false, false, le(s(z1), x0), s(id(x0)), s(id(s(z1))))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Instantiation

                                          ↳ QDP

                                            ↳ Instantiation

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

Q DP problem:
The TRS P consists of the following rules:

IF2(false, z1, s(z2), z3) → IF3(z1, s(z2), z3)
IF3(true, s(x0), s(x1)) → MOD(minus(x0, x1), s(s(x1)))
MOD(s(x0), s(0)) → IF_MOD(false, false, true, s(id(x0)), id(s(0)))
MOD(s(x0), s(s(z1))) → IF_MOD(false, false, le(s(z1), x0), s(id(x0)), s(id(s(z1))))
IF3(true, x0, 0) → MOD(x0, s(0))
IF_MOD(false, y_0, y_1, s(y_2), y_3) → IF2(y_0, y_1, s(y_2), y_3)

The TRS R consists of the following rules:

minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
le(s(x), s(y)) → le(x, y)
id(0) → 0
id(s(x)) → s(id(x))
le(0, y) → true
le(s(x), 0) → false

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
id(0)
id(s(x0))
minus(x0, 0)
minus(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule MOD(s(x0), s(0)) → IF_MOD(false, false, true, s(id(x0)), id(s(0))) at position [4] we obtained the following new rules:

MOD(s(x0), s(0)) → IF_MOD(false, false, true, s(id(x0)), s(id(0)))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Instantiation

                                          ↳ QDP

                                            ↳ Instantiation

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

Q DP problem:
The TRS P consists of the following rules:

IF2(false, z1, s(z2), z3) → IF3(z1, s(z2), z3)
IF3(true, s(x0), s(x1)) → MOD(minus(x0, x1), s(s(x1)))
MOD(s(x0), s(s(z1))) → IF_MOD(false, false, le(s(z1), x0), s(id(x0)), s(id(s(z1))))
IF3(true, x0, 0) → MOD(x0, s(0))
MOD(s(x0), s(0)) → IF_MOD(false, false, true, s(id(x0)), s(id(0)))
IF_MOD(false, y_0, y_1, s(y_2), y_3) → IF2(y_0, y_1, s(y_2), y_3)

The TRS R consists of the following rules:

minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
le(s(x), s(y)) → le(x, y)
id(0) → 0
id(s(x)) → s(id(x))
le(0, y) → true
le(s(x), 0) → false

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
id(0)
id(s(x0))
minus(x0, 0)
minus(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule MOD(s(x0), s(s(z1))) → IF_MOD(false, false, le(s(z1), x0), s(id(x0)), s(id(s(z1)))) at position [4,0] we obtained the following new rules:

MOD(s(x0), s(s(z1))) → IF_MOD(false, false, le(s(z1), x0), s(id(x0)), s(s(id(z1))))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Instantiation

                                          ↳ QDP

                                            ↳ Instantiation

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

Q DP problem:
The TRS P consists of the following rules:

IF2(false, z1, s(z2), z3) → IF3(z1, s(z2), z3)
IF3(true, s(x0), s(x1)) → MOD(minus(x0, x1), s(s(x1)))
IF3(true, x0, 0) → MOD(x0, s(0))
MOD(s(x0), s(s(z1))) → IF_MOD(false, false, le(s(z1), x0), s(id(x0)), s(s(id(z1))))
IF_MOD(false, y_0, y_1, s(y_2), y_3) → IF2(y_0, y_1, s(y_2), y_3)
MOD(s(x0), s(0)) → IF_MOD(false, false, true, s(id(x0)), s(id(0)))

The TRS R consists of the following rules:

minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
le(s(x), s(y)) → le(x, y)
id(0) → 0
id(s(x)) → s(id(x))
le(0, y) → true
le(s(x), 0) → false

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
id(0)
id(s(x0))
minus(x0, 0)
minus(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule MOD(s(x0), s(0)) → IF_MOD(false, false, true, s(id(x0)), s(id(0))) at position [4,0] we obtained the following new rules:

MOD(s(x0), s(0)) → IF_MOD(false, false, true, s(id(x0)), s(0))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Instantiation

                                          ↳ QDP

                                            ↳ Instantiation

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Instantiation

Q DP problem:
The TRS P consists of the following rules:

IF2(false, z1, s(z2), z3) → IF3(z1, s(z2), z3)
IF3(true, s(x0), s(x1)) → MOD(minus(x0, x1), s(s(x1)))
IF3(true, x0, 0) → MOD(x0, s(0))
MOD(s(x0), s(s(z1))) → IF_MOD(false, false, le(s(z1), x0), s(id(x0)), s(s(id(z1))))
MOD(s(x0), s(0)) → IF_MOD(false, false, true, s(id(x0)), s(0))
IF_MOD(false, y_0, y_1, s(y_2), y_3) → IF2(y_0, y_1, s(y_2), y_3)

The TRS R consists of the following rules:

minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
le(s(x), s(y)) → le(x, y)
id(0) → 0
id(s(x)) → s(id(x))
le(0, y) → true
le(s(x), 0) → false

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
id(0)
id(s(x0))
minus(x0, 0)
minus(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule IF3(true, x0, 0) → MOD(x0, s(0)) we obtained the following new rules:

IF3(true, s(z1), 0) → MOD(s(z1), s(0))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Instantiation

                                          ↳ QDP

                                            ↳ Instantiation

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Instantiation

                                                                                                      ↳ QDP

                                                                                                        ↳ Instantiation

Q DP problem:
The TRS P consists of the following rules:

IF2(false, z1, s(z2), z3) → IF3(z1, s(z2), z3)
IF3(true, s(x0), s(x1)) → MOD(minus(x0, x1), s(s(x1)))
IF3(true, s(z1), 0) → MOD(s(z1), s(0))
MOD(s(x0), s(s(z1))) → IF_MOD(false, false, le(s(z1), x0), s(id(x0)), s(s(id(z1))))
MOD(s(x0), s(0)) → IF_MOD(false, false, true, s(id(x0)), s(0))
IF_MOD(false, y_0, y_1, s(y_2), y_3) → IF2(y_0, y_1, s(y_2), y_3)

The TRS R consists of the following rules:

minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
le(s(x), s(y)) → le(x, y)
id(0) → 0
id(s(x)) → s(id(x))
le(0, y) → true
le(s(x), 0) → false

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
id(0)
id(s(x0))
minus(x0, 0)
minus(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule IF_MOD(false, y_0, y_1, s(y_2), y_3) → IF2(y_0, y_1, s(y_2), y_3) we obtained the following new rules:

IF_MOD(false, false, y_0, s(y_1), s(s(y_2))) → IF2(false, y_0, s(y_1), s(s(y_2)))
IF_MOD(false, false, true, s(y_0), s(0)) → IF2(false, true, s(y_0), s(0))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Instantiation

                                          ↳ QDP

                                            ↳ Instantiation

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Instantiation

                                                                                                      ↳ QDP

                                                                                                        ↳ Instantiation

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

Q DP problem:
The TRS P consists of the following rules:

IF2(false, z1, s(z2), z3) → IF3(z1, s(z2), z3)
IF3(true, s(x0), s(x1)) → MOD(minus(x0, x1), s(s(x1)))
IF_MOD(false, false, y_0, s(y_1), s(s(y_2))) → IF2(false, y_0, s(y_1), s(s(y_2)))
IF_MOD(false, false, true, s(y_0), s(0)) → IF2(false, true, s(y_0), s(0))
IF3(true, s(z1), 0) → MOD(s(z1), s(0))
MOD(s(x0), s(s(z1))) → IF_MOD(false, false, le(s(z1), x0), s(id(x0)), s(s(id(z1))))
MOD(s(x0), s(0)) → IF_MOD(false, false, true, s(id(x0)), s(0))

The TRS R consists of the following rules:

minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
le(s(x), s(y)) → le(x, y)
id(0) → 0
id(s(x)) → s(id(x))
le(0, y) → true
le(s(x), 0) → false

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
id(0)
id(s(x0))
minus(x0, 0)
minus(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule IF2(false, z1, s(z2), z3) → IF3(z1, s(z2), z3) we obtained the following new rules:

IF2(false, z0, s(z1), s(s(z2))) → IF3(z0, s(z1), s(s(z2)))
IF2(false, true, s(z0), s(0)) → IF3(true, s(z0), s(0))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Instantiation

                                          ↳ QDP

                                            ↳ Instantiation

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Instantiation

                                                                                                      ↳ QDP

                                                                                                        ↳ Instantiation

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

IF3(true, s(x0), s(x1)) → MOD(minus(x0, x1), s(s(x1)))
IF_MOD(false, false, y_0, s(y_1), s(s(y_2))) → IF2(false, y_0, s(y_1), s(s(y_2)))
IF2(false, z0, s(z1), s(s(z2))) → IF3(z0, s(z1), s(s(z2)))
IF_MOD(false, false, true, s(y_0), s(0)) → IF2(false, true, s(y_0), s(0))
IF3(true, s(z1), 0) → MOD(s(z1), s(0))
MOD(s(x0), s(s(z1))) → IF_MOD(false, false, le(s(z1), x0), s(id(x0)), s(s(id(z1))))
IF2(false, true, s(z0), s(0)) → IF3(true, s(z0), s(0))
MOD(s(x0), s(0)) → IF_MOD(false, false, true, s(id(x0)), s(0))

The TRS R consists of the following rules:

minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
le(s(x), s(y)) → le(x, y)
id(0) → 0
id(s(x)) → s(id(x))
le(0, y) → true
le(s(x), 0) → false

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
id(0)
id(s(x0))
minus(x0, 0)
minus(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 4 less nodes.

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Instantiation

                                          ↳ QDP

                                            ↳ Instantiation

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Instantiation

                                                                                                      ↳ QDP

                                                                                                        ↳ Instantiation

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Instantiation

Q DP problem:
The TRS P consists of the following rules:

IF3(true, s(x0), s(x1)) → MOD(minus(x0, x1), s(s(x1)))
IF_MOD(false, false, y_0, s(y_1), s(s(y_2))) → IF2(false, y_0, s(y_1), s(s(y_2)))
IF2(false, z0, s(z1), s(s(z2))) → IF3(z0, s(z1), s(s(z2)))
MOD(s(x0), s(s(z1))) → IF_MOD(false, false, le(s(z1), x0), s(id(x0)), s(s(id(z1))))

The TRS R consists of the following rules:

minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
le(s(x), s(y)) → le(x, y)
id(0) → 0
id(s(x)) → s(id(x))
le(0, y) → true
le(s(x), 0) → false

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
id(0)
id(s(x0))
minus(x0, 0)
minus(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule IF3(true, s(x0), s(x1)) → MOD(minus(x0, x1), s(s(x1))) we obtained the following new rules:

IF3(true, s(z1), s(s(z2))) → MOD(minus(z1, s(z2)), s(s(s(z2))))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Instantiation

                                          ↳ QDP

                                            ↳ Instantiation

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Instantiation

                                                                                                      ↳ QDP

                                                                                                        ↳ Instantiation

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Instantiation

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Instantiation

Q DP problem:
The TRS P consists of the following rules:

IF_MOD(false, false, y_0, s(y_1), s(s(y_2))) → IF2(false, y_0, s(y_1), s(s(y_2)))
IF2(false, z0, s(z1), s(s(z2))) → IF3(z0, s(z1), s(s(z2)))
MOD(s(x0), s(s(z1))) → IF_MOD(false, false, le(s(z1), x0), s(id(x0)), s(s(id(z1))))
IF3(true, s(z1), s(s(z2))) → MOD(minus(z1, s(z2)), s(s(s(z2))))

The TRS R consists of the following rules:

minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
le(s(x), s(y)) → le(x, y)
id(0) → 0
id(s(x)) → s(id(x))
le(0, y) → true
le(s(x), 0) → false

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
id(0)
id(s(x0))
minus(x0, 0)
minus(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule MOD(s(x0), s(s(z1))) → IF_MOD(false, false, le(s(z1), x0), s(id(x0)), s(s(id(z1)))) we obtained the following new rules:

MOD(s(x0), s(s(s(z1)))) → IF_MOD(false, false, le(s(s(z1)), x0), s(id(x0)), s(s(id(s(z1)))))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Instantiation

                                          ↳ QDP

                                            ↳ Instantiation

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Instantiation

                                                                                                      ↳ QDP

                                                                                                        ↳ Instantiation

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Instantiation

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Instantiation

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

Q DP problem:
The TRS P consists of the following rules:

MOD(s(x0), s(s(s(z1)))) → IF_MOD(false, false, le(s(s(z1)), x0), s(id(x0)), s(s(id(s(z1)))))
IF_MOD(false, false, y_0, s(y_1), s(s(y_2))) → IF2(false, y_0, s(y_1), s(s(y_2)))
IF2(false, z0, s(z1), s(s(z2))) → IF3(z0, s(z1), s(s(z2)))
IF3(true, s(z1), s(s(z2))) → MOD(minus(z1, s(z2)), s(s(s(z2))))

The TRS R consists of the following rules:

minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
le(s(x), s(y)) → le(x, y)
id(0) → 0
id(s(x)) → s(id(x))
le(0, y) → true
le(s(x), 0) → false

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
id(0)
id(s(x0))
minus(x0, 0)
minus(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule MOD(s(x0), s(s(s(z1)))) → IF_MOD(false, false, le(s(s(z1)), x0), s(id(x0)), s(s(id(s(z1))))) at position [4,0,0] we obtained the following new rules:

MOD(s(x0), s(s(s(z1)))) → IF_MOD(false, false, le(s(s(z1)), x0), s(id(x0)), s(s(s(id(z1)))))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Instantiation

                                          ↳ QDP

                                            ↳ Instantiation

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Instantiation

                                                                                                      ↳ QDP

                                                                                                        ↳ Instantiation

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Instantiation

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Instantiation

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ ForwardInstantiation

Q DP problem:
The TRS P consists of the following rules:

IF_MOD(false, false, y_0, s(y_1), s(s(y_2))) → IF2(false, y_0, s(y_1), s(s(y_2)))
IF2(false, z0, s(z1), s(s(z2))) → IF3(z0, s(z1), s(s(z2)))
MOD(s(x0), s(s(s(z1)))) → IF_MOD(false, false, le(s(s(z1)), x0), s(id(x0)), s(s(s(id(z1)))))
IF3(true, s(z1), s(s(z2))) → MOD(minus(z1, s(z2)), s(s(s(z2))))

The TRS R consists of the following rules:

minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
le(s(x), s(y)) → le(x, y)
id(0) → 0
id(s(x)) → s(id(x))
le(0, y) → true
le(s(x), 0) → false

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
id(0)
id(s(x0))
minus(x0, 0)
minus(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [14] the rule IF2(false, z0, s(z1), s(s(z2))) → IF3(z0, s(z1), s(s(z2))) we obtained the following new rules:

IF2(false, true, s(x1), s(s(x2))) → IF3(true, s(x1), s(s(x2)))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Instantiation

                                          ↳ QDP

                                            ↳ Instantiation

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Instantiation

                                                                                                      ↳ QDP

                                                                                                        ↳ Instantiation

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Instantiation

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Instantiation

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ ForwardInstantiation

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ ForwardInstantiation

Q DP problem:
The TRS P consists of the following rules:

IF_MOD(false, false, y_0, s(y_1), s(s(y_2))) → IF2(false, y_0, s(y_1), s(s(y_2)))
IF2(false, true, s(x1), s(s(x2))) → IF3(true, s(x1), s(s(x2)))
MOD(s(x0), s(s(s(z1)))) → IF_MOD(false, false, le(s(s(z1)), x0), s(id(x0)), s(s(s(id(z1)))))
IF3(true, s(z1), s(s(z2))) → MOD(minus(z1, s(z2)), s(s(s(z2))))

The TRS R consists of the following rules:

minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
le(s(x), s(y)) → le(x, y)
id(0) → 0
id(s(x)) → s(id(x))
le(0, y) → true
le(s(x), 0) → false

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
id(0)
id(s(x0))
minus(x0, 0)
minus(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [14] the rule IF_MOD(false, false, y_0, s(y_1), s(s(y_2))) → IF2(false, y_0, s(y_1), s(s(y_2))) we obtained the following new rules:

IF_MOD(false, false, true, s(x1), s(s(x2))) → IF2(false, true, s(x1), s(s(x2)))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Instantiation

                                          ↳ QDP

                                            ↳ Instantiation

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Instantiation

                                                                                                      ↳ QDP

                                                                                                        ↳ Instantiation

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Instantiation

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Instantiation

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ ForwardInstantiation

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ ForwardInstantiation

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Narrowing

Q DP problem:
The TRS P consists of the following rules:

IF_MOD(false, false, true, s(x1), s(s(x2))) → IF2(false, true, s(x1), s(s(x2)))
IF2(false, true, s(x1), s(s(x2))) → IF3(true, s(x1), s(s(x2)))
MOD(s(x0), s(s(s(z1)))) → IF_MOD(false, false, le(s(s(z1)), x0), s(id(x0)), s(s(s(id(z1)))))
IF3(true, s(z1), s(s(z2))) → MOD(minus(z1, s(z2)), s(s(s(z2))))

The TRS R consists of the following rules:

minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
le(s(x), s(y)) → le(x, y)
id(0) → 0
id(s(x)) → s(id(x))
le(0, y) → true
le(s(x), 0) → false

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
id(0)
id(s(x0))
minus(x0, 0)
minus(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MOD(s(x0), s(s(s(z1)))) → IF_MOD(false, false, le(s(s(z1)), x0), s(id(x0)), s(s(s(id(z1))))) at position [2] we obtained the following new rules:

MOD(s(s(x1)), s(s(s(y1)))) → IF_MOD(false, false, le(s(y1), x1), s(id(s(x1))), s(s(s(id(y1)))))
MOD(s(0), s(s(s(y1)))) → IF_MOD(false, false, false, s(id(0)), s(s(s(id(y1)))))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Instantiation

                                          ↳ QDP

                                            ↳ Instantiation

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Instantiation

                                                                                                      ↳ QDP

                                                                                                        ↳ Instantiation

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Instantiation

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Instantiation

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ ForwardInstantiation

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ ForwardInstantiation

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Narrowing

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

IF_MOD(false, false, true, s(x1), s(s(x2))) → IF2(false, true, s(x1), s(s(x2)))
IF2(false, true, s(x1), s(s(x2))) → IF3(true, s(x1), s(s(x2)))
MOD(s(s(x1)), s(s(s(y1)))) → IF_MOD(false, false, le(s(y1), x1), s(id(s(x1))), s(s(s(id(y1)))))
IF3(true, s(z1), s(s(z2))) → MOD(minus(z1, s(z2)), s(s(s(z2))))
MOD(s(0), s(s(s(y1)))) → IF_MOD(false, false, false, s(id(0)), s(s(s(id(y1)))))

The TRS R consists of the following rules:

minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
le(s(x), s(y)) → le(x, y)
id(0) → 0
id(s(x)) → s(id(x))
le(0, y) → true
le(s(x), 0) → false

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
id(0)
id(s(x0))
minus(x0, 0)
minus(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Instantiation

                                          ↳ QDP

                                            ↳ Instantiation

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Instantiation

                                                                                                      ↳ QDP

                                                                                                        ↳ Instantiation

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Instantiation

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Instantiation

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ ForwardInstantiation

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ ForwardInstantiation

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Narrowing

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

Q DP problem:
The TRS P consists of the following rules:

IF_MOD(false, false, true, s(x1), s(s(x2))) → IF2(false, true, s(x1), s(s(x2)))
IF2(false, true, s(x1), s(s(x2))) → IF3(true, s(x1), s(s(x2)))
MOD(s(s(x1)), s(s(s(y1)))) → IF_MOD(false, false, le(s(y1), x1), s(id(s(x1))), s(s(s(id(y1)))))
IF3(true, s(z1), s(s(z2))) → MOD(minus(z1, s(z2)), s(s(s(z2))))

The TRS R consists of the following rules:

minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
le(s(x), s(y)) → le(x, y)
id(0) → 0
id(s(x)) → s(id(x))
le(0, y) → true
le(s(x), 0) → false

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
id(0)
id(s(x0))
minus(x0, 0)
minus(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule MOD(s(s(x1)), s(s(s(y1)))) → IF_MOD(false, false, le(s(y1), x1), s(id(s(x1))), s(s(s(id(y1))))) at position [3,0] we obtained the following new rules:

MOD(s(s(x1)), s(s(s(y1)))) → IF_MOD(false, false, le(s(y1), x1), s(s(id(x1))), s(s(s(id(y1)))))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Instantiation

                                          ↳ QDP

                                            ↳ Instantiation

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Instantiation

                                                                                                      ↳ QDP

                                                                                                        ↳ Instantiation

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Instantiation

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Instantiation

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ ForwardInstantiation

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ ForwardInstantiation

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Narrowing

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ QDPOrderProof

Q DP problem:
The TRS P consists of the following rules:

IF_MOD(false, false, true, s(x1), s(s(x2))) → IF2(false, true, s(x1), s(s(x2)))
IF2(false, true, s(x1), s(s(x2))) → IF3(true, s(x1), s(s(x2)))
MOD(s(s(x1)), s(s(s(y1)))) → IF_MOD(false, false, le(s(y1), x1), s(s(id(x1))), s(s(s(id(y1)))))
IF3(true, s(z1), s(s(z2))) → MOD(minus(z1, s(z2)), s(s(s(z2))))

The TRS R consists of the following rules:

minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
le(s(x), s(y)) → le(x, y)
id(0) → 0
id(s(x)) → s(id(x))
le(0, y) → true
le(s(x), 0) → false

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
id(0)
id(s(x0))
minus(x0, 0)
minus(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MOD(s(s(x1)), s(s(s(y1)))) → IF_MOD(false, false, le(s(y1), x1), s(s(id(x1))), s(s(s(id(y1)))))

The remaining pairs can at least be oriented weakly.

IF_MOD(false, false, true, s(x1), s(s(x2))) → IF2(false, true, s(x1), s(s(x2)))
IF2(false, true, s(x1), s(s(x2))) → IF3(true, s(x1), s(s(x2)))
IF3(true, s(z1), s(s(z2))) → MOD(minus(z1, s(z2)), s(s(s(z2))))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(IF2(x₁, x₂, x₃, x₄)) = x₃
POL(IF3(x₁, x₂, x₃)) = x₂
POL(IF_MOD(x₁, x₂, x₃, x₄, x₅)) = x₄
POL(MOD(x₁, x₂)) = 1 + x₁
POL(false) = 0
POL(id(x₁)) = x₁
POL(le(x₁, x₂)) = 0
POL(minus(x₁, x₂)) = x₁
POL(s(x₁)) = 1 + x₁
POL(true) = 0

The following usable rules [17] were oriented:

id(0) → 0
id(s(x)) → s(id(x))
minus(s(x), s(y)) → minus(x, y)
minus(x, 0) → x

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Rewriting

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ Instantiation

                                          ↳ QDP

                                            ↳ Instantiation

                                              ↳ QDP

                                                ↳ Instantiation

                                                  ↳ QDP

                                                    ↳ UsableRulesProof

                                                      ↳ QDP

                                                        ↳ Rewriting

                                                          ↳ QDP

                                                            ↳ Rewriting

                                                              ↳ QDP

                                                                ↳ UsableRulesProof

                                                                  ↳ QDP

                                                                    ↳ QReductionProof

                                                                      ↳ QDP

                                                                        ↳ Rewriting

                                                                          ↳ QDP

                                                                            ↳ Rewriting

                                                                              ↳ QDP

                                                                                ↳ Rewriting

                                                                                  ↳ QDP

                                                                                    ↳ Rewriting

                                                                                      ↳ QDP

                                                                                        ↳ Rewriting

                                                                                          ↳ QDP

                                                                                            ↳ Rewriting

                                                                                              ↳ QDP

                                                                                                ↳ Rewriting

                                                                                                  ↳ QDP

                                                                                                    ↳ Instantiation

                                                                                                      ↳ QDP

                                                                                                        ↳ Instantiation

                                                                                                          ↳ QDP

                                                                                                            ↳ Instantiation

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Instantiation

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Instantiation

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Rewriting

                                                                                                                              ↳ QDP

                                                                                                                                ↳ ForwardInstantiation

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ ForwardInstantiation

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Narrowing

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Rewriting

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

IF_MOD(false, false, true, s(x1), s(s(x2))) → IF2(false, true, s(x1), s(s(x2)))
IF2(false, true, s(x1), s(s(x2))) → IF3(true, s(x1), s(s(x2)))
IF3(true, s(z1), s(s(z2))) → MOD(minus(z1, s(z2)), s(s(s(z2))))

The TRS R consists of the following rules:

minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
le(s(x), s(y)) → le(x, y)
id(0) → 0
id(s(x)) → s(id(x))
le(0, y) → true
le(s(x), 0) → false

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
id(0)
id(s(x0))
minus(x0, 0)
minus(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 3 less nodes.